Problem 23 - College Algebra

Question

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least one male child.

Step-by-Step Solution

Answer

The probability of selecting a family with at least one male child is \(\frac{7}{8}\) or 0.875.

Step 1: Understanding the Sample Space

The sample space given includes all possibilities for having three children, considering each can be male (M) or female (F), giving a total of 8 outcomes.

Step 2: Identifying Favorable Outcomes

To find the probability of having at least one male child, can count the outcomes that include at least one 'M'. These outcomes are: MMM, MMF, MFM, MFF, FMM, FMF, FFM. So there are 7 favorable outcomes.

Step 3: Calculating the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of outcomes. So, the probability of having at least one male child is \(\frac{7}{8} = 0.875\).

Key Concepts

Understanding the Sample Space

In probability, the sample space is the set of all possible outcomes of an experiment. For example, when selecting a family with three children, each child can either be male (M) or female (F). This means we can have different combinations of M and F, which are considered all possible outcomes.

Here, we list those combinations:

• MMM
• MMF
• MFM
• MFF
• FMM
• FMF
• FFM
• FFF

Each string represents a possible gender combination for the three children. There are a total of 8 outcomes, making the sample space complete for this situation.

Identifying Favorable Outcomes

Favorable outcomes are the specific results that satisfy the condition we are interested in. In this case, we're looking for families with at least one male child (M).

To find these, we examine our sample space:

• MMM
• MMF
• MFM
• MFF
• FMM
• FMF
• FFM

Out of the 8 possible outcomes, 7 include at least one 'M'. These are our favorable outcomes. Identifying these helps us determine the probability of the event in question.

Equally Likely Outcomes

In probability, equally likely outcomes mean that each outcome in the sample space has the same chance of occurring. When selecting a family with three children, each combination of M and F is equally probable.

This assumption simplifies our calculations because it allows us to apply a straightforward probability formula. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes.

For this problem, the probability of selecting a family with at least one male child is given by:\[\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{7}{8} = 0.875\]Understanding equally likely outcomes ensures that our probability calculations are accurate and meaningful.